372 APPENDIX: ELEMENTS OF LINEAR ALGEBRA

(A.5.3) Now let G be another finite-dimensional vector space over K, and
let (ck)l^k^p be a basis of G. Let u: E->F and v : F-*G be two linear
mappings and let w = v ° u : E -» G. Suppose that the matrix M(u) of u
with respect to (at) and (bj), and the matrix M(v) of v with respect to (4y)
and (cjt) are known, and let us calculate the matrix M(w) of w with respect
to fa) and fo). If Af(w) = (o^), Jlf(i>) = (fly), M(w) = (yft£), then by definition
we have

p / m \ m m / p \

y c*l

p / m \ m m / p

= Z ?«<* = o E % &/ = E o/i *>(&/) = Z <ty( Z A

fc=l V/=l / 7=1 7=1 \fc=l

from which it follows that
(A.5.3.1)

x /? matrix Af(w) is said to be the product of the/? x m matrix M(v) and
the m x n matrix M(u), and we write

M(v o u) = M(v)M(u).

Thus the product of two matrices is calculated by " multiplying the rows of the
first by the columns of the second."

Having given these definitions, it is of course possible to translate into
matrix language most of the results we have established for linear mappings.
We shall not stop to do this: indeed, in practice it is almost always advanta-
geous, when presented with a problem of matrix algebra, to reformulate it
in the language, so much more flexible and appropriate, of linear mappings.
For example, there is no simple interpretation, in terms of matrices, of the
essential concepts of the kernel and the image of a linear mapping.

6. MULTILINEAR MAPPINGS. DETERMINANTS

(A.6.1) Let El9...,Er and F be r + 1 vector spaces over a field K. A
mapping u : Ej x E2 x • • • x Er -> F is said to be r-linear if it is "linear in
each of its arguments " : that is to say, for each j = 1, 2, . . . , r and each choice
of elements a,- e E,- (J ^ z), the partial mapping

. . , at.ly xt , at+i, ...
of Et into F is linear. This implies in particular that
wfa, ...,ai_1,0,e H' is a hyperplane.